Long-time asymptotics for the massive Thirring model

The massive Thirring model (MTM) was introduced in 1958 by the Austrian physicist Walter Thirring in the context of relativistic quantum field theory. It describes the self-interaction of a Dirac field in one space dimension. From the analytical point of view, this system of non-linear partial differential equations is of special interest, because it has a representation in terms of a Lax pair, consisting of two linear operators $L$ and $A$. Thanks to the Lax pair, the MTM admits an exact solution by the inverse scattering transform (IST). Since the dependence of $L$ and $A$ on the spectral parameter $\lambda$ is singular at the origin and at infinity, the IST cannot be defined for initial data of low regularity as straightforward as it is done for other equations, the NLS equation for instance. One key ingredient of the present thesis is to transform the known Lax pair to two equivalent Lax pairs: one is suitable for the spectral parameter at the origin and the other one is suitable at infinity. Using the equivalent operators the direct scattering transform is developed for an optimal $L^2$-based Sobolev space. The inverse scattering map is then formulated in terms of two Riemann--Hilbert problems whose solvability is proven. As it is also known from other nonlinear dispersive equations one can create \emph{solitons} for the MTM. These special solutions are waves that move at constant speed and do not change in shape. They can refuse to disperse only because of the presence of the nonlinearity in the equation. It is relatively simple to characterize solitons, based on their scattering data. Using suitable Riemann-Hilbert techniques it is possible to analyse the interaction of two (or more) solitons. Furthermore, it can be shown precisely that each soliton will eventually enter the light cone $\set{|t|>|x|}$. Using the so-called \db--method (nonlinear steepest descent) we show that outside the light cone any solution (not only solitons) converges to zero with a rate of $\sim|t|^{-3/4}$. Inside the light-cone there are basically two different possibilities. Assuming that the initial data is free of solitons we use the \db--method and some well-known model Riemann--Hilbert--problems to show that the solution of the MTM scatters to a linear solution modulo phase correction. This linear solution can be computed explicitly from the scattering data and its amplitude decays with a rate of $\sim|t|^{-1/2}$. The second possibility is that the initial data contains finitely many solitons. Then, as the main result of the thesis, we prove that any solution breaks up into finitely many single solitons that travel at different speeds and thus, diverge. The remainder term is $\mathcal{O}(|t|^{-1/2})$.\medskip\\ Summarizing, the present thesis provides an analytical proof of the soliton resolution conjecture for the MTM. This result also implies the asymptotic stability of solitons

The derivative NLS equation: global existence with solitons

We prove the global existence result for the derivative NLS equation in the case when the initial datum includes a finite number of solitons. This is achieved by an application of the Backlund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann-Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently. A major difficulty in the proof is to show invertibility of the Backlund transformation acting on weighted Sobolev spaces